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Let $V$ be an irreducible finite dimensional real representation of a real finite dimensional Lie algebra $\mathfrak{g}$. From Schur's Lemma, what is $Hom_\mathfrak{g}(V,V)$ or $End_\mathfrak{g}(V,V)$?

I am getting confused with the terminology employed in the literature. Do we have a (real?) (non?) associative (division?) ring/algebra/algebraic extension?

Many thanks in advance for this clarification.

Nina
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  • Schur lemma tells you that $\mathrm{End}_{\mathfrak{g}}(V)$ is a division ring. Since it is obviously finite dimensional over $\mathbb{R}$, it is $\mathbb{R}$, $\mathbb{C}$ or $\mathbb{H}$. –  Mar 19 '15 at 11:02
  • Many thanks for your answers. –  Mar 19 '15 at 11:05

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Note that $End_{\mathfrak{g}}(V)$ is an associative subalgebra of $End(V)$; and it is a real division algebra by Schur's lemma. The real division algebras of finite dimension are $\mathbb{R}, \mathbb{C}, \mathbb{H}, \mathbb{O}$, but $\mathbb{O}$ is not associative.

Dietrich Burde
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